Optimal Carbon Abatement in a Stochastic Equilibrium Model with Climate Change

Christoph Hambela, Holger Kraftb,*, Eduardo Schwartzc

a,bGoethe University Frankfurt, Faculty of Economics and Business Administration, Theodor-W.-Adorno-Platz 3, 60323 Frankfurt am Main, Germany.c UCLA Anderson School of Management, Los Angeles.

This version: July 13, 2017

Abstract

This paper studies a dynamic stochastic general equilibrium model involving climate change. Our frame-work allows for feedback effects on the temperature dynamics. We are able to match estimates of futuretemperature distributions provided in the fifth assessment report of the IPCC (2014). We compare twoapproaches to capture damaging effects of temperature on output (level vs. growth rate impact) andcombine them with two degrees of severity of this damage (Nordhaus vs. Weitzman calibration). It turnsout that the choice of the damage function is crucial to answer the question of how much the distinctionbetween level and growth rate impact matters. The social cost of carbon is similar for frameworks withlevel or growth rate impact if the potential damages of global warming are moderate. On the otherhand, they are more than twice as large for a growth rate impact if damages are presumably severe. Wealso study the effect of varying risk aversion and elasticity of intertemporal substitution on our results.If damages are moderate for high temperatures, risk aversion only matters when climate change has alevel impact on output, but the effects are relatively small. By contrast, the elasticity of intertemporalsubstitution has a significant effect for both level and growth rate impact. If damages are potentiallysevere for high temperatures, then the results also become sensitive to risk aversion for both damagespecifications.

JEL-Classification: D81, Q5, Q54

Keywords: Climate change economics, carbon abatement, social cost of carbon, GDP growth, stochasticdifferential utility

∗ Corresponding author. Phone: +49 69 798 33700.

E-mail addresses: christoph.hambel@finance.uni-frankfurt.de (Christoph Hambel),holgerkraft@finance.uni-frankfurt.de (Holger Kraft),eduardo.schwartz@anderson.ucla.edu (Eduardo Schwartz)

Acknowledgements: We thank Christian Gollier, Frederick van der Ploeg, Armon Rezai, and RichardTol for helpful comments and suggestions. We also thank participants of the 23rd Annual Conference ofthe European Association of Environmental and Resource Economists (EAERE 2017), the FinanceUCConference, Santiago, Chile, the Santiago Finance Workshop, Chile, the Frankfurt-Mannheim MacroWorkshop, and the joint seminar of the Humboldt University Berlin and ESMT for their comments andsuggestions. All remaining errors are of course our own. Holger Kraft and Christoph Hambel gratefullyacknowledge financial support by Deutsche Forschungsgemeinschaft (DFG). the Center of ExcellenceSAFE, funded by the State of Hessen initiative for research LOEWE.

Optimal Carbon Abatement in a Stochastic Equilibrium Model with Climate Change

Abstract

This paper studies a dynamic stochastic general equilibrium model involving climate change. Our frame-work allows for feedback effects on the temperature dynamics. We are able to match estimates of futuretemperature distributions provided in the fifth assessment report of the IPCC (2014). We compare twoapproaches to capture damaging effects of temperature on output (level vs. growth rate impact) andcombine them with two degrees of severity of this damage (Nordhaus vs. Weitzman calibration). It turnsout that the choice of the damage function is crucial to answer the question of how much the distinctionbetween level and growth rate impact matters. The social cost of carbon is similar for frameworks withlevel or growth rate impact if the potential damages of global warming are moderate. On the otherhand, they are more than twice as large for a growth rate impact if damages are presumably severe. Wealso study the effect of varying risk aversion and elasticity of intertemporal substitution on our results.If damages are moderate for high temperatures, risk aversion only matters when climate change has alevel impact on output, but the effects are relatively small. By contrast, the elasticity of intertemporalsubstitution has a significant effect for both level and growth rate impact. If damages are potentiallysevere for high temperatures, then the results also become sensitive to risk aversion for both damagespecifications.

JEL-Classification: D81, Q5, Q54

Keywords: Climate change economics, carbon abatement, social cost of carbon, GDP growth, stochasticdifferential utility

1 Introduction

Our paper proposes a stochastic optimization-based general equilibrium model for the optimal

abatement policy and optimal consumption. In contrast to most of the literature we allow for

random evolutions of the key variables such as CO2 concentration, global temperature and world

GDP. We determine the optimal abatement policy and study this policy across different future

scenarios for several model specifications. We provide detailed calibrations where we simulta-

neously match two decisive climate-sensitivity measures (TCR, ECS) which play an important

role in the report of the IPCC (2014). In particular, we analyze the implications of different

assumptions about the impact of climate change on output (level or growth) if damages of global

warming are more or less pronounced (Nordhaus and Sztorc (2013) or Weitzman (2012)). We

identify situations where the difference between growth rate and level impact matters and also

study the effect of risk aversion and elasticity of intertemporal substitution (EIS).

If climate damages are moderate as in Nordhaus and Sztorc (2013), median results such as size

of the social cost of carbon (SCC) over the next 100 years are similar for level and growth

rate impact. However, a level impact induces a high variation in optimal emissions, whereas

a growth rate impact leads to a less state-dependent abatement policy. The risk aversion has

a smaller effect than the elasticity of intertemporal substitution (EIS) and is negligible for

the growth rate impact. If climate damages are severe for high temperatures as in Weitzman

(2012), there are pronounced differences between level and growth rate impact. Now, a growth

rate impact leads to significantly higher SCC and induces a more state-dependent abatement

policy. Furthermore, the results are sensitive to both the choice of risk aversion and EIS. In any

case, abatement activities are more drastic, and optimal emissions are more volatile.

Notice that the social cost of carbon – an indicator of damage done by emitting carbon dioxide

– is tightly varying with the optimal abatement policy. It is thus a stochastic process that

increases in temperature and that depends on risk aversion and EIS in a similar way as the

optimal abatement policy. Depending on the specification the optimal policy may or may not

vary significantly with temperature. For instance, for a growth rate effect the policies are less

state-dependent for moderate damages, but are crucially state-dependent for severe damages.

Therefore, predications of deterministic models (e.g. DICE model) are also more or less accurate

depending on the true nature of the effect. This is however an empirical question that is beyond

the scope of our study.

Our paper is related to several studies using integrated assessment models (IAM):1 Firstly, the

DICE model (Dynamic Integrated Model of Climate and the Economy) is the most common

framework to study optimal carbon abatement. It combines a Ramsey-type model for capital

1IAMs can broadly be divided into two classes: welfare optimization models which choose an optimal abatementpolicy and simulation models that renounce an optimization routine and rather evaluate specific policy scenar-ios. Such a framework combines knowledge from different areas of science to an unified model that describesinteractions between greenhouse gas emissions, the climate system and the economy.

1

allocation with deterministic dynamics of emissions, carbon dioxide and global temperature.

Notice that it is formulated in a deterministic setting, see for example Nordhaus (1992, 2008),

Nordhaus and Sztorc (2013).2 Kelly and Kolstad (1999) and Kelly and Tan (2015) extend this

model and allow the decision maker to learn about the unknown relation between greenhouse

gas emissions and temperature. In frameworks with recursive utility, Crost and Traeger (2014),

Jensen and Traeger (2014), and Ackerman et al. (2013) analyze versions where one component

is assumed to be stochastic. In contrast to our paper, Crost and Traeger (2014) and Jensen and

Traeger (2014) do not allow for stochastic temperature dynamics, but consider uncertainties in

economic growth and the damage function. Ackerman et al. (2013) introduce transitory uncer-

tainty of the climate sensitivity parameter into the DICE model. These studies indicate that

risk aversion has a smaller effect on the social cost of carbon than the elasticity of intertemporal

substitution. We confirm the earlier results for moderate climate damages and show that risk

aversion can have a crucial effect if damages are severe for high temperatures. Cai et al. (2015)

study a stochastic generalization referred to as DSICE model. Their approach is computation-

ally involved since it is based on high-dimensional Markov chains. However, both carbon and

temperature dynamics are deterministic and the model only involves a level impact on economic

growth induced by climate change (as all variants of the DICE model).

By contrast, we suggest a flexible IAM that does not fall into the class of DICE models.3 All

components are genuinely stochastic. In particular, it involves parsimonious stochastic dynamics

for the world temperature. We can thus offer a way to simultaneously calibrate two decisive

climate-sensitivity measures (TCR, ECS). Transient climate response (TCR) measures the total

increase in average global temperature at the date of carbon dioxide doubling. Equilibrium

climate sensitivity (ECS) refers to the change in global temperature that would result from a

sustained doubling of the atmospheric carbon dioxide concentration after the climate system

will have found its new equilibrium. Notice that the fifth assessment report of the IPCC (2014)

provides estimates for the distribution of these measures in the future. In particular, it gives

empirical evidence about non-negligible risks for extreme climate changes. In other words, future

temperatures are supposed to be fat-tailed and right-skewed. Our approach allows us to match

moments beyond the first and second moment. This gives us the opportunity to study the

effects of fat-tailed and skewed temperature distributions and to capture some of the inherent

uncertainty of the problem.4

Other related literature includes Bansal and Ochoa (2011), Dell et al. (2009, 2012) and Burke

et al. (2015) who provide empirical evidence that temperature negatively affects the growth rate

of output rather than its level (as in the DICE approach). Pindyck (2011, 2014) studies the effect

2When we refer to DICE in this article, we mean the DICE-2013R-version that is presented in Nordhaus andSztorc (2013).

3To simplify comparisons with the existing literature, we calibrate the growth rate in the benchmark case suchthat our economic model closely matches the business-as-usual (BAU) evolution of GDP in DICE.

4See, e.g., the remarks of Nordhaus (2008) on the uncertainty of the problem.

2

Economic Model‐ Gross Domestic Product‐ Green Technology‐ Abatement Cost‐ Economic Shocks

Equilibrium Maximize global welfare by choosing an optimal abatement policy and optimal consumption

Damage Process: Translate damages in the ecosystem in reduced economic growth

Abatement Policy:  Expenditures for green technologies are costly, but reduce emissions

Climate Change Process: Emissions yield an increase

in global temperature

Carbon Dioxide Model‐ Carbon Dioxide Emissions‐ Carbon Dioxide Concentration‐ Natural Sinks, Carbon Shocks

Climate Model‐ Global Temperature‐ Climate Shocks ‐ Feedback Effects / Fat Tails

Figure 1: Building Blocks of the Model.

of a growth rate impact in an endowment economy. He solves a static instead of a dynamic

optimization problem and calculates the so-called willingness to pay. This is the fraction of

consumption that is necessary to keep global warming below some target temperature, e.g. 3◦C.

However, he abstracts from carbon dioxide emissions and abatement costs. Dietz and Stern

(2015) study a stochastic version of DICE that is plagued by persistent impacts on economic

growth and involves a fat-tailed ECS. Using a Monte-Carlo approach, they provide a solution

where decisions are formed before the first period and are not revised. Moore and Diaz (2015)

study the effect of growth rates impacts in a deterministic two-region version of DICE and find

that a growth rate impact warrants stringent mitigation policy. Finally, similar as in our paper,

Pindyck (2012) studies the difference between level and growth rate effect, but in the stylized

setting of Pindyck (2011, 2014).

As in most of the above-mentioned papers, the starting point for our economic analysis of

climate change is an integrated assessment model. Consequently, our model consists of three

components: carbon dioxide model, climate model, and economic model. Section 2 describes

these components and characterizes the equilibrium of the economy. Section 3 calibrates all

model components. Section 4 presents our benchmark results. Additional robustness checks can

be found in Section 5. Section 6 concludes. An Appendix contains additional material.

2 Model Setup

This section presents the model setup and describes its equilibrium. Figure 1 depicts the three

building blocks of our framework (carbon dioxide model, climate model, and economic model).

The carbon dioxide model keeps track of the carbon dioxide concentration in the atmosphere.

This concentration increases by anthropological and also non-man made carbon dioxide shocks

3

and it decreases since natural sinks such as oceans absorb carbon dioxide. Society can control

anthropological carbon dioxide emissions by choosing an abatement strategy which reduces the

current (business-as-usual) emissions. These efforts are costly.

The climate model measures the average world temperature and its departure from the pre-

industrial level. Empirically, there is a (noisy) positive relation between carbon dioxide concen-

tration and world temperature. Our temperature process captures this relation and allows for

possible feedback effects.

The economic model describes the dynamics of global GDP (syn. output) in a stylized production

economy. In our benchmark setting, global warming has a negative influence on economic

growth, i.e. on the drift of global GDP. Alternatively, we also study a framework with a level

impact as in DICE. Society can only indirectly mitigate this damaging effect by choosing the

above mentioned abatement strategy. This is the link of the economic model to the emission

model, which completes the circle.

Society (syn. mankind or decision maker) chooses optimal consumption and an optimal abate-

ment strategy whose costs contemporaneously reduce economic growth. The remaining part of

output must be invested so that an equilibrium materializes.

2.1 Carbon Dioxide Model

The average pre-industrial concentration of carbon dioxide in the atmosphere is denoted by Y PI.

The total current concentration of carbon dioxide in the atmosphere is given by

Y Σt = YPI + Yt, (1)

where Yt denotes the amount of atmospheric carbon dioxide that is caused by human activities,

i.e. the part of atmospheric carbon dioxide that exceeds the pre-industrial concentration. Its

dynamics are assumed to be

dYt = Yt [(µy(t)− αt)dt+ σydW yt ] . (2)

We refer to (2) as carbon dioxide dynamics or process. Here W y = (W yt )t≥0 is a standard

Brownian motion that models unexpected shocks on the carbon dioxide concentration. These

could be the result of environmental shocks such as volcano eruptions or earthquakes, but they

can also be man-made. The volatility of these shocks σy is assumed to be constant. Atmospheric

carbon dioxide increases with an expected growth rate of µy that models the current growth path

of the carbon dioxide concentration. In other words, µy is the growth rate if society does not

take additional actions to reduce carbon dioxide emissions. We thus refer to µy as the business-

as-usual drift of the carbon dioxide process. Notice that it also involves all past policies which

have been implemented to reduce carbon dioxide emissions.

4

Society can however pursue new policies to reduce emissions. We refer to such an additional

effort as an abatement strategy α = (αt)t≥0. In other words, the abatement policy α models how

additional actions change the expected growth of the carbon dioxide concentration, i.e. these are

abatement policies beyond business-as-usual (BAU). By definition, this differential abatement

policy has been zero in the past (αt = 0 for all t < 0). If no abatement policy is chosen and

society sticks to BAU, we also use the notation Y BAU instead of Y .

Our dynamics of the carbon concentration Y are formulated in terms of the abatement policy

α. However, we are also interested in the resulting CO2 emissions. To back out the implied

CO2 emissions that are consistent with (2), we now consider alternative dynamics of Y where

– up to environmental shocks – the change in Y is expressed as the difference between CO2

emissions and the amount of carbon absorbed by natural sinks. Formally, if et denotes the

time-t anthropological carbon dioxide emissions, then we obtain

dYt = ζeeαt dt− δy(Xt)Ytdt+ YtσydW

yt , (3)

dXt = δy(Xt)Ytdt, (4)

where ζe is a factor converting emissions into concentrations.5 The variable Xt measures the

total quantity of atmospheric carbon dioxide that has already been absorbed by natural sinks.

The function δy models the decay rate of carbon dioxide, i.e. the speed at which carbon dioxide

is absorbed from the atmosphere. We assume δy to decrease in X, i.e. the capacity of natural

sinks declines with the quantity of carbon that has already been absorbed. Equation (3) can

be considered as an ecological budget constraint : The total change in carbon dioxide is (up

to environmental shocks) the difference between anthropological emissions and natural carbon

sequestration.

The dynamics (2) and (3) can be interpreted as a system of two equations with the two unknowns

dY and eα. By equating (2) and (3), we can solve for the anthropological emissions of carbon

dioxide (short: emissions) that are consistent with both dynamics:

eαt =Ytζe

[µy(t) + δy(Xt)− αt] . (5)

Equation (5) provides the relation between the abatement strategy and the anthropological

emissions under that strategy. We use the notation eBAUt for business-as-usual emissions (α = 0).

Finally, we define the so-called emission control rate as

εα = (eBAU − eα)/eBAU = 1− YtζeeBAUt

(µy(t) + δy(Xt)− αt). (6)

5Carbon dioxide emissions are measured in gigatons (GtCO2), whereas concentrations are measured in partsper million (ppm).

5

This quantity denotes the fraction of abated carbon dioxide emissions compared to BAU. Equiv-

alently, it is the percentage of carbon dioxide emissions that is prevented from entering the

atmosphere if the abatement policy α is implemented. As in the DICE model, we assume that

the emission control rate εα is between 0 and 1. The assumption εα ≥ 0 excludes strategies thatlead to emissions beyond BAU. On the other hand, the assumption εα ≤ 1 implies that emissionscannot be negative, which might only be possible if there are major technological breakthroughs

(e.g. direct carbon removal (DCR)).

Notice that the restriction εα ≤ 1 yields to the following upper bound on the abatement policy

αt ≤ µy(t) + δy(Xt) (7)

i.e. technological restrictions prevent society from implementing very high abatement policies.

This constraint makes it harder to make up for opportunities that have been missed in the past.6

2.2 Climate Model

We assume that the average global increase in temperature from its pre-industrial level is given

by the dynamics7

dTt =Ytητ

Y Σt

(µy(t)− αt

)dt+

Ytστ

Y Σt

(ρyτdW

yt +

√1− ρ2yτdW τt

)+ θτ (Tt−)dN

τt . (8)

We refer to (8) as global warming process. The parameter ητ is a constant relating the change in

global temperature to changes in carbon dioxide concentration. The Brownian motions W τ and

W y are independent. The diffusion parameter στ is assumed to be constant. Furthermore, Nτ =

(N τt )t≥0 is a self-exciting process whose jump intensity πτ (Tt) and jump size θτ (Tt) can depend

on Tt itself. There is empirical evidence that the distribution of future temperature changes

is right-skewed (see IPCC (2014)). One reason for this is that there might be delayed climate

feedback loops triggered by increases in global temperature. This line of argument suggests that

the temperature dynamics involve a self-exciting jump process whose jump intensity and jump

size depend on the temperature itself. Intuitively, this means that an increase in temperature

makes future increases both more likely and potentially more severe. Therefore, a self-exciting

process captures the idea of feedback loops and at the same time allows for calibrating the

skewness of the distribution of future temperature changes.

6If it were really possible to actively remove carbon dioxide from the atmosphere (direct carbon removal), thennegative CO2 emissions would be feasible. As in the DICE model, we do not allow for negative emissions in ourbenchmark calibration. However, our results are robust to this assumption. In robustness checks not reportedhere, we have assumed that the emission control rate is restricted to εαt ∈ [0, 1.2], where εαt > 1 involves negativeemissions. On a time scale of 100 years, our median main results however hardly change. Only on extreme paths,society implements more stringent abatement policies leading to negative emissions.

7Appendix A provides a motivation for these climate dynamics and describes the technical details.

6

2.3 Economic Model

2.3.1 GDP Growth

Following Pindyck and Wang (2013) we assume that real GDP (syn. output) is generated by a

production technology that is linear in capital (AK-technology) and involves capital adjustment

costs.8 In our model, global warming has a negative impact on GDP. We assume that real GDP

is

Ct = ĈtKαt Dt (9)

where Kαt is a factor that captures the costs resulting from implementing the abatement policyα. Section 2.3.2 below describes how Kαt is related to the abatement policy. Dt is a damagefactor that scales down output in response to climate change. Section 2.3.3 discusses the form of

the damage factor Dt in greater detail. In particular, we study a damage factor that influences

the growth rate of GDP. Notice that Ĉt denotes real GDP in the absence of global warming

(shadow GDP). Put differently, Ĉt is what GDP would be if there were no climate change. Its

dynamics are given by

dĈt = Ĉt [g(t, χt)dt+ σc(ρcydWyt + ρ̂cτdW

τt + ρ̂cdW

ct )] , (10)

where W c = (W ct )t≥0 is a third Brownian motion that is independent of Wy, W τ and N τ . The

volatility σc of the economic shocks is assumed to be constant. Output is correlated with carbon

concentration and temperature via ρcy and ρcτ . Standard arguments then lead to the following

specifications:9

ρ̂cτ =ρcτ − ρcyρyτ√

1− ρ2cy, ρ̂c =

√1− ρ2cy − ρ̂2cτ .

Shadow GDP evolves with an expected growth rate g(t, χ). Here, χ denotes the consumption

rate, i.e. the fraction of GDP that is consumed. In our framework, the expected growth rate of

shadow GDP has the following form (see Appendix B for more details)

g(t, χ) = A(1− χ)− δk(t)−1

2ϑ(1− χ)2, (11)

where the capital-output ratio A is constant (AK-technology). The deterministic function δk(t)

describes the trend of the capital stock dynamics. It captures the effects of depreciation, but

also the growth of human capital. The parameter ϑ measures the costs of installing new capital.

This specification ensures that capital adjustment costs are homogenous in capital and convex

8Formally, output is Ct = AKt, where A > 0 is a constant modeling productivity and K is the only factorof production. K is the total stock of capital, i.e. it includes physical capital, but also human capital and firm-based intangible capital. Appendix B describes how the GDP dynamics can be derived from such a productiontechnology.

9Formally, this is a Cholesky decomposition.

7

in investment and abatement. Such an assumption is widely used in the literature (see, e.g.,

Hayashi (1982) and Jermann (1998), Pindyck and Wang (2013)).

2.3.2 Abatement Costs

Output C can be used for consumption C, investments I or abatement expenditures Aα,

Ct = Ct + It +Aαt . (12)

Implementing an abatement policy α can be considered as spending money on more carbon effi-

cient technologies that reduce carbon dioxide emissions (green technologies). The expenditures

necessary to implement the policy α are the so-called abatement expenditures denoted by Aαt .To keep the model tractable, we measure and calibrate these expenditures relative to output.10

As can be seen from (9), we model the negative effects of an abatement policy α on output via a

cost process Kαt that scales down output as a response to abatement expenditures. We assumethat the cost process Kαt is of finite variation and can be represented by a sufficiently smoothcost function κ that depends on time and the emission control rate:

Kαt = exp(−∫ t

0κ(s, εαs )ds

). (13)

Therefore, implementing α reduces economic growth over the small interval [t, t+dt] by κ(t, εαt )dt.

We refer to κ(t, εαt ) as the instantaneous growth effect of the abatement expenditures Aα. Putdifferently, κ(t, εαt ) models the forgone economic growth resulting from spending money on

abatement. Appendix B shows that κ(t, εαt ) can be calculated if the abatement expenditures

Aα, productivity A, and output C are known:

κ(t, εαt ) = AAαtCt. (14)

Using relation (6), we can rewrite κ in terms of time t, carbon concentration Y and abatement

policy α. Therefore, we also use the notation κ(t, Yt, αt) instead of κ(t, εαt ).

2.3.3 Impact of Global Warming

This paper studies two approaches of how to model economic damages induced by climate

change. First, we consider a widely used approach assuming that current temperatures directly

affect the level of GDP (see, e.g., Nordhaus (2008), Tol (2002), Hope (2006)). Second, we

analyze a framework that models damages as a negative effect on the growth rate of GDP, which

is suggested by empirical evidence.

10This is standard in the literature (e.g. DICE). Relaxing this assumption would lead to an additional statevariable.

8

Level Impact The standard approach assumes that global warming has a direct impact on the

level of GDP which is captured by a sufficiently smooth damage function D(T ) with D(0) = 1,

limT→∞D(T ) = 0. Here T denotes the temperature anomaly. Thus, real GDP at time t is

Ct = ĈtKαt D(Tt). (15)

Notice that the damage function only depends on the current temperature anomaly. Therefore,

in this specification only the current level of GDP is affected, but not its growth rate. We

thus refer to such a damage specification as level impact. As Pindyck (2012, 2013) points out,

a level impact has the unrealistic drawback that damages from global warming are reversible.

For instance, if global temperatures increase heavily over 100 years, but then come back to the

present level, GDP would fully recover without any permanent losses. Therefore, the impact of

global warming as modeled by (15) is transitory.

Growth Rate Impact There is a lot of empirical evidence that higher temperatures affect

growth rates of GDP rather than just its level (see, e.g., Dell et al. (2009, 2012)). In line with

these findings, Pindyck (2012, 2013) argues that many damages induced by climate change are

likely to be permanent, rather than transitory (e.g. destruction of ecosystems, sea level rise, or

heat-related fatalities). Therefore, it is relevant to study the effects of a growth rate impact as

well. In this case, we assume that real GDP at time t is given by

Ct = ĈtKαt exp(−ζd

∫ t0T ks ds

)(16)

where ζd and k are positive parameters that relate temperature increase to loss of economic

growth. Apparently, in such a framework the growth rate depends on the whole path of the

temperature anomaly rather than just on the current value as in (15). It thus captures long-

lasting damages from climate change that are difficult or impossible to reverse (at least in human

timescales). Combining (10) and (16) yields the following dynamics of real GDP

dCt = Ct

[(g(t, χ)− ζdT kt )dt+ σc(ρcydW

yt + ρ̂cτdW

τt + ρ̂cdW

ct )− κ(t, εαt )dt

], (17)

i.e. the expected growth rate g(t, χt)− ζdT kt is negatively affected by current temperatures. Thedynamics (17) are the reason why we refer to this damage specification as growth rate impact.11

2.4 Equilibrium

It is well-known that for a decision maker with CRRA utility changing the degree of relative risk

aversion has at first sight a counterintuitive effect: The abatement policy is less stringent if risk

11Notice that for both specifications the distribution of future climate damages is known. Therefore, the DismalTheorem of Weitzman (2009) does not apply.

9

aversion increases.12 In order to resolve this puzzle and to disentangle relative risk aversion from

elasticity of intertemporal substitution, we follow Crost and Traeger (2014), Jensen and Traeger

(2014) and Ackerman et al. (2013) and assume the decision maker’s preferences to be of Epstein-

Zin type. This allows us to analyze the effects of varying EIS and risk aversion separately. The

society’s time-t utility index Jα,χt associated with a given abatement-consumption strategy (α, χ)

over the infinite planning horizon [0,∞) is thus recursively defined by

Jα,χt = Et[∫ ∞

tf(Cα,χs , Jα,χs )ds

], (18)

where Cα,χ = χCα,χ denotes consumption. Following Duffie and Epstein (1992) the aggregatorfunction f is given by the continuous-time Epstein-Zin aggregator

f(C, J) =

δθJ

[(C

[(1−γ)J ]1

1−γ

)1− 1ψ − 1

], ψ 6= 1

δ(1− γ)J log(

C[(1−γ)J ]

11−γ

), ψ = 1

(19)

with θ = 1−γ1−1/ψ . The parameter γ > 1 measures the degree of relative risk aversion, ψ > 0

reflects the elasticity of intertemporal substitution (EIS), and δ > 0 denotes the time-preference

rate.13 For θ = 1 (or equivalently ψ = 1/γ), the preferences simplify to standard time-additive

CRRA utility with utility function u(c) = 11−γ c1−γ . For θ < 1 (i.e. ψ > 1/γ) the agent prefers

early resolution of uncertainty and is eager to learn outcomes of random events before they

occur. On the other hand, if θ > 1 (i.e. ψ < 1/γ) the agent prefers late resolution of uncertainty.

Notice that although recursive utility allows to disentangle risk aversion from EIS, it does not

allow to disentangle prudence from the other two parameters as well. Following Kimball and

Weil (2009) prudence is given by γ(1 + ψ). Therefore, risk aversion and EIS affect prudence in

a linear way. We will discuss the impact of prudence in the robustness section where we vary

risk aversion and EIS separately.

The decision maker chooses an admissible abatement-consumption strategy (α, χ) in order to

maximize his utility index Jα,χt at any point in time t ∈ [0,∞). An admissible strategy mustensure that output, consumption, investment and abatement expenditures remain positive, i.e.

Ct, Ct, It,Aαt ≥ 0 for all t ≥ 0. Furthermore, the abatement policy must satisfy (7) and lead toa positive emission control rate. The class of all admissible abatement-consumption strategies

12Pindyck (2013) explains this fact as follows: For a higher level of risk aversion, the marginal utility ofconsumption declines faster. However, consumption is expected to grow and consequently utility from futureconsumption decreases with risk aversion. For a higher level of risk aversion society thus implements a lessstringent abatement policy leading to higher emissions and a higher global temperature.

13Although empirical evidence suggests that γ > 1 is the reasonable specification for the index of relative riskaversion, it is also possible to define aggregator functions for γ ∈ [0, 1].

10

at time t is denoted by At. The indirect utility function is given by

J(t, c, x, y, τ) = sup(α,χ)∈At

{Jα,χt | Ct = c,Xt = x, Yt = y, Tt = τ} (20)

We solve the utility maximization problem (20) by applying the dynamic programming principle.

Details of the HJB equation and the solution method are presented in Appendix F.

2.5 Social Cost of Carbon

Our model can be used to calculate the social cost of carbon (SCC). Following Nordhaus and

Sztorc (2013), Traeger (2014) and others, we define the social cost of carbon as the marginal rate

of substitution between carbon dioxide emission and GDP. Formally, the social cost of carbon

is given by

SCCt = −∂Jt∂et

/ ∂Jt∂Ct

. (21)

Intuitively, the social cost of carbon measures the increase in current GDP that is required to

compensate for economic damages caused by an marginal increase of time-t emissions. Therefore,

SCC can be interpreted as an optimal carbon tax, i.e. the tax that compensates for the negative

external effects from burning carbon. More details on how SCC is calculated can be found in

Appendix F.

3 Calibration

This section provides a detailed calibration of all model components. Table 1 summarizes the

calibration results and serves as our benchmark calibration. This calibration assumes a growth

rate effect of climate change. We choose the year 2015 as starting point of our model (t = 0).14

3.1 Preferences

In order to disentangle risk aversion from elasticity of intertemporal substitution, we use recursive

preferences. In the literature, there is no consensus on how to choose γ and ψ.15 Many studies

that incorporate recursive utility in an IAM choose γ = 10 and ψ in the range between 0.5 and

1.5 (see, e.g., Ackerman et al. (2013), Crost and Traeger (2014), Jensen and Traeger (2014) and

Cai et al. (2015)). We follow that literature and choose ψ = 1 as the benchmark value for EIS.

14Since DICE starts in 2010 and evolves in steps of 5 years, this assumption simplifies comparisons.15Bansal and Yaron (2004) and Vissing-Joergensen and Attanasio (2003) combine equity and consumption data

and estimate an EIS of 1.5 and a risk aversion in the range between 8 and 10. On the other hand, Hall (1988),Campbell (1999), Vissing-Joergenen (2002) estimate an EIS well below one.

11

Carbon Dioxide ModelY PI Pre-industrial carbon dioxide concentration 280Y0 Initial excess carbon dioxide concentration 121ζe Conversion factor 0.1278σy Carbon dioxide volatility 0.0078

Climate ModelT0 Current global warming 0.9ητ Temperature scaling parameter 2.592στ Temperature volatility 0.1ρyτ CO2/temperature correlation 0.04

Economic ModelC0 Initial GDP (trillion US-$) 75.8A Productivity 0.113ϑ Adjustment cost parameter 0.372σc GDP volatility 0.0162ρcτ GDP/temperature correlation 0ρcy GDP/CO2 correlation 0.29ζd Damage scaling parameter 0.00026

Preferencesδ Time-preference rate 0.015γ Relative risk aversion 10ψ Elasticity of Intertemporal Substitution 1

Table 1: Benchmark Calibration. This table summarizes the parameters of the benchmark calibra-tion which is described in Section 3.

The time-preference rate is δ = 0.015, which is a standard assumption in the IAM literature

(see, e.g., the recent version of the DICE model by Nordhaus and Sztorc (2013)). In robustness

checks, we vary these parameters and study their effects on our results.

3.2 Carbon Dioxide Model

The fifth assessment report of the IPCC (2014) provides four stylized climate scenarios depending

on the future evolution of greenhouse gas emissions referred to as representative concentration

pathways (RCPs). The RCP 8.5 scenario is characterized by high CO2 emissions where the

atmospheric concentration is supposed to stabilize at a high level in the second half of the 23th

century.16 Consequently, the RCP 8.5 data is well-suited to serve as the average BAU scenario

for CO2 emissions and concentrations. Notice that all RCPs are deterministic, i.e. they can only

be used to calibrate averages. Therefore, we use historical data to estimate the randomness of

the carbon dioxide concentration.

Carbon Dioxide Dynamics To calibrate (1) and (2), we fix the pre-industrial carbon dioxide

concentration at Y PI = 280 ppm, which is a common assumption in the literature. Furthermore,

in the year 2015 (t = 0) the carbon dioxide concentration was 401 ppm, which implies Y0 = 121

16The data is available at http://tntcat.iiasa.ac.at/RcpDb

12

2100 2200 23000

0.5

1

1.5

2

2.5

Year

(a) Drift [%]

2100 2200 2300400

600

800

1000

1200

1400

1600

1800

2000

Year

(b) Concentration [ppm]

2100 2200 23000

20

40

60

80

100

120

Year

(c) Emissions [GtCO2]

Figure 2: Calibration of the Carbon Dioxide Model. The crosses in Graph (a) depict the implieddrift of the evolution of atmospheric carbon dioxide in the RCP 8.5 scenario. The solid line is ourcalibration of µy(t). The crosses in Graph (b) depict the evolution of atmospheric carbon dioxide in theRCP 8.5 scenario. The solid line shows our calibration to that data. The crosses in Graph (c) depictthe emission prognosis in the RCP 8.5 scenario. The solid line shows our calibration to that data and anextension until 2300.

ppm as starting value for the carbon dioxide process (2). Then we calibrate the drift µy(t) such

that the drift of the average BAU evolution (i.e. α = 0 and σy = 0 in (2)) is close to the drift of

the RCP 8.5 scenario that is marked by crosses in Graph (a) of Figure 2.17 Obviously, RCP 8.5

assumes three different regimes. For the first 40 years, the drift is virtually flat at a level close

to the historical trend. Then the drift falls to zero over the next 200 years where it remains

afterwards. This functional form of the drift rate can be captured in the following way:

µy(t) = 0.022 1{t

Ecological Budget Constraint In a second step, we calibrate the decay rate of carbon

dioxide δy(Xt) such that the model-implied carbon dioxide emissions (5) match the RCP 8.5

emissions (crosses in Graph (c) of Figure 2). The main issue here is that RCP 8.5 provides con-

centration data until 2300, but emission data only until 2100. We thus perform our calibration

in two steps: First, we use both concentration and emission data until 2100 and determine the

functional form of δy. Here we fix the conversion factor at ζe = 0.1278ppm

GtCO2, which converts

emissions into concentrations (see, e.g., IPCC (2014) and the references therein). Then we use

this functional form and the concentration data to extrapolate the emissions until 2300.

As can be seen from Graph (b), the concentration of RCP 8.5 has an inflection point around

2100 and remains flat after the year 2240. Consequently, the emissions of RCP 8.5 must be

hump-shaped. Since these emissions level off around 2100 in the data (crosses in Graph (c)), it

is reasonable to expect a turning point around that year or shortly after, although - as noted

above - RCP 8.5 is silent about emissions after the year 2100.19 This is exactly what our

extrapolation yields.

The solid line in Graph (c) depicts the fit to that data and our BAU-emission forecast until 2300.

It turns out that the following functional form of the decay rate of carbon dioxide matches the

data well:

δy(x) = aδe−(x−bδcδ

)2

where we estimate aδ = 0.0176, bδ = −27.63, cδ = 314.8 (R2 > 99%). Appendix C describes thetechnical details. Notice that the presumed evolution of BAU emissions beyond 2100 is similar

to the baseline evolution in DICE. For instance, in the year 2200 DICE predicts 59GtCO2, which

is close to the estimate of 54GtCO2 in our model.

3.3 Climate Model

The calibration of the global warming process (8) is divided into two steps: First, we calibrate the

direct impact of the carbon dioxide concentration on global warming (captured by the continuous

part of the model). The drift is calibrated using historical data, whereas the estimate of the

volatility involves data on the transient climate response (TCR). In a second step, we calibrate

the jump size and jump intensity such that the model can generate the above mentioned feedback

effects. Here we use data on the equilibrium climate sensitivity (ECS).

Direct Impact: Drift and Volatility To estimate the parameter ητ in the drift of the

process, we use historical data on carbon dioxide concentration and global warming.20 Notice

19Therefore, we can merely extrapolate the emissions from 2100 onwards. It is however obvious that concen-tration can only flatten out if emissions eventually decrease and reach a low level where natural sinks can absorball emissions such that concentration does not increase any more.

20Source: United Kingdom’s national weather service. Global warming data available athttp://www.metoffice.gov.uk/.

14

300 350 400−0.2

0

0.2

0.4

0.6

0.8

1

(a) Temperature Anomaly [°C]

CO2 Concentration [ppm]

1 2 30

0.02

0.04

0.06

0.08

0.1

0.12

0.14(b) Transient Climate Response

Temperature Anomaly [°C]0 5 10

0

0.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18(c) Equilibrium Climate Sensitivity

Temperature Anomaly [°C]

Figure 3: Calibration of the Climate Model. The crosses in Graph (a) depict pairs of empiricalglobal warming and atmospheric carbon dioxide concentration. The solid line depicts the regression curve(23). The estimated parameters of the fitted curve is ητ = 2.592. Graph (b) shows a histogram of thesimulated transient climate response. Graph (c) depicts a histogram of the equilibrium climate sensitivity.The histograms are based on a simulation of 1 million sample paths.

that the starting point for our model of the global warming dynamics is (24) in Appendix A.

Therefore, we estimate ητ by running a linear regression of global warming data on log-carbon

dioxide data. Put differently, we calculate

minητ

N∑i=1

[Ti − ητ log

(Y ΣiY PI

)]2. (23)

Here Ti denotes the temperature above the pre-industrial level and YΣi denotes the carbon

dioxide concentration at time ti. Our estimation yields ητ = 2.592. The linear model performs

well with R2 > 0.8. Graph (a) of Figure 3 depicts the data and the estimate. We also use

that data in order to estimate the correlation between carbon dioxide and global warming. We

obtain a correlation parameter ρyτ = 0.04.

To calibrate the diffusion coefficient στ of (8), we use data on a measure called the transient

climate response (TCR). TCR measures the total increase in average global temperature at

the date of carbon dioxide doubling, t2× = inft{t ≥ 0 | Yt = Y PI}. The data comes fromCMIP5.21 They simulate the future climate dynamics and obtain a multimodel mean (as well

as median) of about E[TCR] = 1.8◦ and a 90% confidence interval of [1.2◦C, 2.4◦C]. This pointstowards an approximately symmetric distribution of TCR, which is in line with our Brownian

assumption. Further, notice that our above estimate of ητ leads to a total temperature increase

21CMIP5 refers to Coupled Model Intercomparison Project Phase 5. See http://cmip-pcmdi.llnl.gov/cmip5/for further information.

15

of about ητ log(2) = 1.797 at the relevant date t2× for TCR. This is also in line with the CMIP5

estimate. Therefore, we are left with finding στ , which we achieve by using the information

about the confidence interval. The 95%-quantile is 1.65 standard deviations above the mean.

This implies a standard deviation of σTCR = 0.6◦C/1.65 = 0.364◦C. We choose the volatility

parameter στ such that our model fits the distribution of TCR at the time when carbon dioxide

is supposed to double. For this purpose, we estimate the doubling time t2× via Monte Carlo

simulation: We sample 1 million uncontrolled carbon dioxide paths to determine the time of

carbon dioxide doubling. Then, we simulate 1 million global warming paths and choose the

diffusion parameter such that the simulated distribution of TCR matches the above mentioned

quantiles at the time of carbon dioxide doubling (see Graph (b) of Figure 3).22 On average,

doubling occurs in 2055. As a result of the calibration, we estimate στ = 0.1 and a small

correlation of about ρyτ = 0.04.

Feedback Effects: Jumps In a second step, we calibrate the jump intensity and size using

IPCC estimates for the equilibrium climate sensitivity (ECS). ECS refers to the change in global

temperature that would result from a sustained doubling of the atmospheric carbon dioxide

concentration after the climate system will have found its new equilibrium. This process is

presumably affected by feedback effects kicking in after the temperature has increased signif-

icantly (e.g. the date related to TCR). Since the jump part in our model captures feedback

effects, we use ECS data to estimate the corresponding parameters. Unfortunately, there is

no consensus distribution for ECS because finding a new equilibrium might take hundreds of

years. Summarizing more than 20 scientific studies, the IPCC (2014) however states that ECS

is “likely” in the range of 1.5◦C to 4.5◦C with a most likely value of about 3◦C.23 Additionally,

there is a probability of 5 to 10% that doubling the carbon dioxide concentration leads to an

increase in global temperature of more than 6◦C, while its extremely unlikely (i.e. less than 5%)

that temperature increase is below 1◦C. These numbers suggest that ECS has a right-skewed

distribution which can be generated by jumps.

We assume that the climate system will find its new equilibrium 100 years after the carbon

dioxide concentration will have doubled. We choose a functional form and an appropriate

parametrization for the jump size and jump magnitude such that we can reproduce the above

mentioned mean and quantiles of ECR by running Monte Carlo simulations. Furthermore, we

perform the calibration in such a way that the constructed distribution for TCR is preserved.

The latter is achieved by allowing for very small negative jumps when the temperature increase

is still low. We thus choose the following parametrization of the climate shock intensity and

22Here we set the jump part equal to zero such that the results are not driven by warming feedback effects. Seealso the definition of ECS in the next section.

23In the language of IPCC, the word “likely” means with a probability higher than 67%.

16

Specification Calibrated with respect to Parametrization

Level Impact(L-N) Nordhaus and Sztorc (2013) DN (T ) = 1

1+0.00266T2

(L-W) Weitzman (2012) DW (T ) = 11+(T/20.64)2+(T/6.081)6.754

Growth Impact(G-N) Nordhaus and Sztorc (2013) ζNd = 0.00026, k = 1(G-W) Weitzman (2012) ζWd = 0.000075, k = 3.25

Table 2: Damage Specifications. The table summarizes the four different damage specifications thatare studied in this paper.

magnitude:

πτ (τ) =

(0.95

1 + 2.8e−0.3325τ− 0.25

)+,

θτ (τ) = −0.0029τ2 + 0.0568τ − 0.0577

Notice that we calibrate the jump intensity such that πτ (τ) = 0 for all τ ≤ 0, i.e. there are nofeedback effects if the global temperature is at or below its pre-industrial level. The simulated

ECS distribution is depicted in Graph (c) of Figure 3.

3.4 Economic Model

We calibrate the growth rate (11) such that our economic model closely matches the evolution

of GDP growth in the DICE model. Additionally, we chose the abatement cost function from

DICE and derive the functional form of κ. The technical details can be found in Appendix D.

In order to analyze the impact of warming, we consider a set of possible specifications. The

standard approach in the literature assumes that warming has a direct impact on the level of

GDP via a sufficiently smooth damage function D(T ) with D(0) = 1. Thus, GDP at time t is

Ct = ĈtD(Tt), where Ĉ denotes GDP in the absence of global warming (shadow GDP). There

is however empirical evidence that rather the growth rate of GDP than the level is affected by

global warming., e.g. Dell et al. (2009, 2012). To compare the effects of different damage types,

we implement our model with four different specifications for the impact of warming. Table 2

summarizes these specifications.

Level Impact The standard damage function in DICE is inverse quadratic. Nordhaus and

Sztorc (2013) use the parametrization

DN (T ) =1

1 + 0.00266T 2,

17

which we refer to as (L-N) specification. They calibrate the damage function to temperature

increases between 0◦C to 3◦C. They acknowledge that adjustments might be needed in case of

higher warming. Weitzman (2012) proposes an alternative damage function that is based on an

expert panel study involving 52 experts on climate economics. His damage function is designed

to capture tipping point effects for very high temperature increases:

DW (T ) =1

1 + (T/20.64)2 + (T/6.081)6.754,

which we refer to as (L-W) specification. The two damage functions are very close for tem-

peratures in the range between 0◦C and 3◦C. From 3◦C onwards, the losses start to deviate

significantly. For instance, for a temperature increase of 6◦C, Nordhaus’ damage function DN

predicts a GDP loss of 9.2% percent, while Weitzman’s specification DW generates a loss of

approximately 50% of GDP.

Growth Rate Impact To compare the effects of level and growth rate impacts, we first

calibrate the growth rate impact such that the GDP dynamics are close to those resulting from

a Nordhaus’ level damage (L-N). In (17) we set k = 1. Furthermore, we choose the damage

parameter ζNd = 0.00026 such that the average GDP losses in the year 2100 coincide for both

specifications. Formally, using (15) and (16) the following equation implicitly determines ζNd

E[e−ζ

Nd

∫ t0 Tsds+σcŴ

ct

]= E

[eσcŴ

ctDN (Tt)

],

where Ŵ ct = ρcyWyt + ρ̂cτW

τt + ρ̂cW

ct and t denotes the year 2100. We refer to the resulting

specification as (G-N). Notice that this parameter is in line with the calibration of Pindyck

(2014). Similarly, we calibrate the growth rate impact (G-W) such that the GDP dynamics

are close to those resulting from a Weitzmans’ level damage (L-W). This yields k = 3.25 and

ζWd = 0.000075.

4 Main Results

This section presents our main results for the model introduced in Section 2. In particular,

we determine the optimal abatement policy, its costs, the evolution of real GDP as well as the

evolution of the carbon dioxide concentration and global average temperature changes over the

next 100 years. Unless otherwise stated, we use our benchmark calibration from Section 3 that

is summarized in Table 1.

18

2020 2040 2060 2080 2100

200

400

600

(a) GDP [trillion US−$]

Year2020 2040 2060 2080 2100

200

400

600

(b) GDP [trillion US−$]

Year

2020 2040 2060 2080 2100

400

600

800

(c) CO2 Concentration [ppm]

Year2020 2040 2060 2080 2100

400

600

800

(d) CO2 Concentration [ppm]

Year

2020 2040 2060 2080 21000

2

4

(e) Global Warming [°C]

Year2020 2040 2060 2080 2100

0

2

4

(f) Global Warming [°C]

Year

2020 2040 2060 2080 21000

20406080

100120

(g) Emissions [GtCO2], Control Rate [%]

Year

0

50

100

2020 2040 2060 2080 21000

20406080

100120

(h) Emissions [GtCO2], Control Rate [%]

Year

0

50

100

2020 2040 2060 2080 21000

100

200

300

(i) Social Cost of Carbon [$/GtCO2]

Year2020 2040 2060 2080 2100

0

100

200

300

(j) Social Cost of Carbon [$/GtCO2]

Year

Figure 4: Results for the Nordhaus Calibration. Based on the calibration of Section 3, the graphsdepict our results for the level impact (L-N) (left column) and the growth rate impact (G-N) (rightcolumn). Optimal paths are depicted by solid lines and BAU paths by dotted lines. Dashed lines show5% and 95% quantiles of the optimal solution. Graphs (a) and (b) deptict the evolution of world GDP,(c) and (d) the carbon dioxide concentration, (e) and (f) changes in global temperature, (g) and (h)carbon dioxide emissions and the median optimal emission control rate (dash-dotted line), (i) and (j) thesocial cost of carbon.

19

Model 2015 2035 2055 2075 2095 2115 2150 2200

(G-N) GDP [trillion $] 75.8 138.9 228.9 345.4 483.7 637.1 941.8 1435.6SCC [$/tCO2] 11.12 21.75 50.67 102.52 171.21 225.10 254.12 353.25Abatement Expenditures [trillion $] 0.01 0.11 0.59 2.02 4.58 6.87 7.11 5.72Emission Control Rate 0.12 0.24 0.41 0.61 0.82 0.95 1 1Temperature rise [◦C] 0.9 1.3 1.7 2.1 2.4 2.5 2.7 3.1

(L-N) GDP [trillion $] 75.8 139.3 230.4 348.7 490.2 652.7 988.0 1558.3SCC [$/tCO2] 10.63 24.23 58.37 116.84 183.03 221.77 254.70 376.68Abatement Expenditures [trillion $] 0.01 0.12 0.73 2.41 4.98 6.69 6.99 6.00Emission Control Rate 0.12 0.25 0.44 0.65 0.83 0.93 1 1Temperature rise [◦C] 0.9 1.3 1.7 2.0 2.3 2.4 2.5 2.9

Table 3: Median Results for the Nordhaus Calibration. The table reports the median evolutionof selected variables for the growth rate (G-N) and level (L-N) impact.

4.1 Level vs. Growth Rate Impact for the Nordhaus Calibration

Table 3 and Figure 4 compare the evolutions of key state variables for the growth and level

damage specifications (G-N) and (L-N). Both models behave similarly until the end of this

century. This is not surprising as we calibrate the growth rate impact (G-N) such that the

BAU evolution of world GDP until 2100 it is close to the one in (L-N). However, there are two

main differences between these specifications. First, although the optimally controlled outputs

in models (G-N) and (L-N) are similar until 2115, they diverge significantly in later years such as

2200 where the median output is almost 9% smaller in the model with growth impact. Second,

the variation of global temperature in (G-N) is much higher than in (L-N), while the variability

of emissions and concentrations is lower.

Notice that for a level impact damages are directly related to the current temperature, whereas

for a growth rate impact damages depend on the whole temperature path so that the weight

of the current temperature is much smaller and the damaging effects of high temperatures are

delayed. Therefore, the abatement policy in (L-N) is slightly more stringent on average, but

far more stringent for high temperatures. This implies a higher variability in carbon dioxide

emissions and concentrations, but a lower variability in global temperature compared to (G-N).

It also leads to a higher median output for (L-N), since more rigorous abatement policies tend

to avoid economic damages more effectively.

Our analysis confirms and extends the results in Pindyck (2012). He shows in a static model

that the willingness to pay24 for keeping global warming below a certain threshold is higher for

level damages than for growth damages, a finding that is in line with our results. However,

Pindyck (2012) also states that there are no substantial differences between the two models.

Our findings challenge this conclusion. First, output levels are significantly different in the year

2200, which is reported in Table 3. Second, the optimal emission path depends strongly on both

24The willingness to pay is defined as the percentage of output that society is willing to sacrifice to keep thetemperatures below a specified threshold.

20

Model 2015 2035 2055 2075 2095 2115 2150 2200

(G-W) GDP [trillion $] 75.8 138.1 223.4 330.3 459.4 610.4 921.1 1451.1SCC [$/tCO2] 42.86 92.55 145.93 172.82 188.20 198.44 219.77 333.00Abatement Expenditures [trillion $] 0.14 0.99 3.06 4.59 5.37 5.63 5.58 4.89Emission Control Rate 0.30 0.53 0.75 0.83 0.87 0.89 0.92 0.96Temperature rise [◦C] 0.9 1.1 1.2 1.3 1.4 1.4 1.4 1.4

(L-W) GDP [trillion $] 75.8 139.2 229.4 343.4 480.6 642.0 975.3 1551.0SCC [$/tCO2] 18.07 42.40 93.19 152.92 189.00 207.09 222.51 346.33Abatement Expenditures [trillion $] 0.03 0.29 1.50 3.69 5.28 5.93 5.46 4.94Emission Control Rate 0.16 0.35 0.57 0.76 0.85 0.89 0.89 0.93Temperature rise [◦C] 0.9 1.2 1.5 1.7 1.8 1.9 2.0 2.4

Table 4: Median Results for the Weitzman Calibration. The table reports the median evolutionof selected variables for the growth rate (G-W) and level (L-W) impact.

the current state of the climate system and the damage specification. For instance by 2095, the

95% quantile of temperature is 3.1 (2.6) ◦C in the model with growth (level) impact leading to

optimal carbon dioxide emissions of 19 (0) GtCO2. This implies that the choice of the damage

specification (growth rate or level impact) can have a significant effect, in particular for extreme

paths.

4.2 Level vs. Growth Rate Impact for the Weitzman Calibration

We now consider the specifications (L-W) and (G-W), which are described in Section 3.4. Table 4

and Figure 5 show our corresponding findings. The graphs of Figure 5 on the left-hand (right-

hand) side depict the results for the level (growth rate) impact. To avoid the potentially severe

consequences of global warming, society keeps temperature low and in a narrow confidence band,

which can be seen in Graphs (e) and (f). In turn, this leads to abatement strategies that are

more sensitive to changes in current temperature. Therefore, most of the damaging effects of

climate change can potentially be avoided resulting in steady economic growth (see Graphs (a)

and (b)). From the end of the century onwards, the BAU paths of GDP are significantly lower

than the optimally controlled paths.

Although in both scenarios society acts more rigorously than in the previous case, there are

quantitative differences between the level impact (L-W) and the growth rate impact (G-W) that

are also qualitatively different from our previous results on the Nordhaus calibration. For the

growth rate impact, SCC is initially 42.86 and thus more than twice as high as for the level

impact where it is 18.07. This implies more rigorous abatement activities in (G-W) than in (L-

W). Therefore, the temperature increase is significantly smaller. Surprisingly, now the growth

rate impact involves a higher SCC. This can intuitively be explained by the attitude of an agent

with recursive preferences towards changes in the drift of his endowment stream. The long-run

risk literature (see, e.g., Bansal and Yaron (2004)) documents that this type of agents is very

sensitive to persistent changes of the growth rate. Whereas in the Nordhaus calibration the

21

2020 2040 2060 2080 2100

200

400

600

(a) GDP [trillion US−$]

Year2020 2040 2060 2080 2100

200

400

600

(b) GDP [trillion US−$]

Year

2020 2040 2060 2080 2100

400

500

600

(c) CO2 Concentration [ppm]

Year2020 2040 2060 2080 2100

400

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600

(d) CO2 Concentration [ppm]

Year

2020 2040 2060 2080 21000

1

2

3(e) Global Warming [°C]

Year2020 2040 2060 2080 2100

0

1

2

3(f) Global Warming [°C]

Year

2020 2040 2060 2080 21000

20406080

100120

(g) Emissions [GtCO2], Control Rate [%]

Year

0

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100

2020 2040 2060 2080 21000

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100120

(h) Emissions [GtCO2], Control Rate [%]

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0

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200

300

(i) Social Cost of Carbon [$/tCO2]

Year2020 2040 2060 2080 2100

0

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300

(j) Social Cost of Carbon [$/tCO2]

Year

Figure 5: Weitzman Damage Specification. The graphs depict our results for the level impact (L-W) (left column) and the growth rate impact (G-W) (right column). Optimal paths are depicted by solidlines and BAU paths by dotted lines. Dashed lines show 5% and 95% quantiles of the optimal solution.Graphs (a) and (b) show the evolution of world GDP, (c) and (d) the carbon dioxide concentration inthe atmosphere, (e) and (f) median changes in global temperature, (g) and (h) carbon dioxide emissionsand the median optimal emission control rate (dash-dotted line), (i) and (j) the social cost of carbon.

22

ψ γ

Nordhaus Calibration 1 2 5 10 150.5 5.58 (5.75) 5.61 (5.80) 5.72 (5.98) 5.93 (6.34) 6.16 (6.79)1 10.80 (9.25) 10.83 (9.38) 10.94 (9.82) 11.12 (10.63) 11.29 (11.59)2 16.16 (12.55) 16.16 (12.71) 16.13 (13.24) 16.05 (14.24) 15.90 (15.41)

Weitzman Calibration 1 2 5 10 150.5 10.54 (6.98) 11.81 (7.26) 16.50 (8.31) 18.24 (10.83) 21.06 (12.54)1 18.73 (11.74) 19.69 (12.23) 24.21 (14.04) 42.86 (18.07) 72.44 (23.71)2 24.58 (15.41) 25.08 (15.98) 26.81 (18.07) 51.14 (22.73) 89.93 (29.05)

Table 5: Sensitivity Analysis of SCC for Risk Aversion and EIS. The table shows SCC [$/tCO2]in 2015 for different values of γ and ψ. The numbers in front of the brackets are the results for the growthrate impact. The numbers in brackets are the results for the corresponding level impact.

effect on the growth rate is apparently too moderate, this property has a significant influence in

the Weitzman calibration.

5 Robustness Checks

This section presents robustness checks for elasticity of intertemporal substitution, risk aversion,

and diffusion parameters. We also compare our findings to the results in the DICE model.

5.1 Preference Parameters

Optimal Abatement and SCC We first consider the effect of varying the elasticity of in-

tertemporal substitution, ψ ∈ {0.5, 1, 2}. Table 5 reports the results for the social cost of carbonand shows a strong dependence on EIS.25 For a high level of EIS, society is willing to accept less

smooth consumption streams. Consequently, it implements a more rigorous abatement policy

raising SCC. The opposite is true for a low level of EIS. These results hold for both level and

growth rate impact regardless of the calibration of the damages.

The effect of varying the degree of relative risk aversion depends on the damage specification

and calibration. If damages are moderate for high temperatures (Nordhaus calibration), risk

aversion is negligible in a model with a growth rate impact (G-N) and slightly more pronounced

with a level impact (L-N). Nevertheless, the effects are relatively small. These results are in

line with the findings of Ackerman et al. (2013) and Crost and Traeger (2014) that risk aversion

has a much smaller effect than EIS on the optimal abatement decision and in turn on SCC.26

25This is in line with the findings of Cai et al. (2015), Crost and Traeger (2014), Jensen and Traeger (2014)and Bansal et al. (2014).

26Crost and Traeger (2014) point out that most integrated assessment models are formulated for a CRRAdecision maker with ψ = 1/γ. Since risk aversion plays an inferior role for the social cost of carbon and theoptimal abatement policy, it is important to calibrate the entangled preference parameters to match EIS, ratherthan risk aversion. Especially for deterministic models, where risk aversion is in fact irrelevant, this might leadto significant changes in the optimal abatement policies.

23

ψ

Nordhaus Calibration χ I/C0.5 [75.4%, 81.9%] (75.4%, 81.9%) [18.1%, 24.2%] (18.0%, 23.7%)1 [75.0%, 75.0%] (75.0%, 75.0%) [24.0%, 25.0%] (23.8%, 25.0%)2 [72.2%, 74.7%] (72.2%, 74.4%) [24.5%, 27.7%] (24.2%, 27.8%)

Weitzman Calibration χ I/C0.5 [75.6%, 81.6%] (75.6%, 82.0%) [17.3%, 23.9%] (18.0%, 24.2%)1 [75.0%, 75.0%] (75.0%, 75.0%) [21.2%, 25.0%] (22.3%, 25.0%)2 [72.2%, 75.1%] (72.2%, 74.6%) [23.3%, 27.7%] (24.1%, 27.8%)

Table 6: Sensitivity Analysis of Consumption and Investment for EIS. The table shows therange of optimal consumption and investment (as fraction of output) for different values of ψ. Thenumbers in box brackets are the results for the growth rate impact. The numbers in curved brackets arethe results for the corresponding level impact.

However, if damages are potentially severe for high temperatures (Weitzman calibration), the

results become sensitive to the choice of risk aversion for both damage specifications. Now, SCC

and optimal abatement policy increase with risk aversion.27

To summarize, abatement expenditures lead to steeper consumption streams (less consumption

today, potentially more consumption in the future) and thus the EIS has a first-order effect. On

the other hand, risk aversion or prudence become only relevant if the consequences of postponing

abatement are severe and significantly state-dependent as in (L-W) and (G-W).

Optimal Consumption and Investment For unit EIS, the optimal consumption rates are

constant. Lemma F.1 shows that for non-unit EIS the optimal consumption rate becomes

state-dependent. Table 6 summarizes the effects of varying EIS on optimal consumption and

investment, both expressed as a fraction of output.

We find that for ψ > 1, the optimal consumption rates are smaller than for unit EIS. Addition-

ally to the more stringent abatement policy, society also installs more new capital via higher

investment rates. Therefore, the gross growth rate of output is higher for ψ > 1. This confirms

our intuition that with higher EIS, society accepts less smooth consumption streams, while the

opposite is true for ψ < 1.28

27Our results also suggest that prudence, which is given by γ(1 + ψ) (see Kimball and Weil (2009)), has asecond-order effect as well. This is because prudence is affected similarly by risk aversion and EIS. If prudencehad a significant effect on our results, then varying γ should also lead to significant changes, but this is only truewhen the consequences of postponing abatement are potentially disastrous.

28Notice that for our benchmark choice of unit EIS, Section 3.4 calibrates ϑ = 0.372 in order to match aconsumption rate of 75%. If we choose ϑ to be 0.32(0.4) for an EIS of 0.5(2), then the consumption rate is inthe range of 72%(74%) and 79%(76%), which is well in line with the historical range of 72% and 78%. Moreimportantly, SCC for the different choices of ϑ are almost identical.

24

σc 0 0.0081 0.0162 0.0243Nordhaus Calibration 11.10 (10.61) 11.11 (10.62) 11.12 (10.63) 11.13 (10.64)Weitzman Calibration 42.84 (18.05) 42.85 (18.06) 42.86 (18.07) 42.86 (18.09)

στ 0 0.05 0.1 0.15Nordhaus Calibration 10.14 (7.44) 10.41 (8.26) 11.12 (10.63) 11.75 (13.81)Weitzman Calibration 15.93 (8,67) 19.95 (10.88) 42.86 (18.07) 69.87 (33.03)

Table 7: SCC for Different Volatility Parameters. The table compares SCC [$/tCO2] for differentvolatility parameters for the four damage specifications. The results of the level specifications are inbrackets.

5.2 Influence of Diffusive Shocks and Feedback Effects

Diffusive Shocks Table 7 shows how SCC in the year 2015 changes if the diffusion parameters

of output and temperature, σc and στ , are varied. It turns out that the volatility σc of economic

shocks has a negligible effect on the current SCC. On the other hand, the effect of στ is significant,

since high variation in temperature amplifies the risk of ending up in a feedback loop during which

temperature increases heavily. This is because the jump intensity increases in temperature.

Therefore, society tries to avoid feedback loops by implementing a more rigorous abatement

policy. Table 7 reports SCC for the four damage specifications. It can also be seen that SCC is

more sensitive for the level impact.

Stochastic Feedback Effects We now analyze the impact of disregarding the stochastic

feedback effects, i.e. πτ (τ) = θτ (τ) = 0. To obtain an expected equilibrium climate sensitivity

of 3◦C, we now choose ητ = 4.33. Notice that this specification can match the first two mo-

ments of ECS, but it cannot generate a fat-tailed climate sensitivity. For the Nordhaus damage

specifications, SCC reduces from 11.12 (10.63) to 8.90 (4.81), where the number in brackets are

the results for the level impact (L-N). Similar, for the Weitzman specifications, SCC in the year

2015 decreases from 42.86 (18.07) to 18.97 (5.34). We thus conclude that fat-tailed distributed

climate dynamics induce a higher social cost of carbon and higher optimal abatement. The effect

is more pronounced for level impacts where a climate feedback loop has potentially disastrous

direct impacts on the economy.

5.3 Comparison with DICE

This subsection compares our benchmark results with those obtained in the DICE version of

Nordhaus and Sztorc (2013). In particular, we compare the optimal social cost of carbon to

Nordhaus’ calculations. Nordhaus estimates the social cost of carbon in 2015 to be 19.6 dollars

(expressed in 2005-dollars per ton of carbon dioxide). He uses a CRRA utility function with

γ = 1.45 implying an EIS of ψ = 1/γ. By contrast, we use recursive preferences with γ = 10 and

ψ = 1. The starting value of the social cost of carbon in our model is lower than estimated in

25

AbatementPolicy 2015 2035 2055 2075 2095 2115 2150 2200

Optimal GDP [trillion $] (5% quantile) 75.8 124.1 195.4 284.7 386.8 501.4 724.7 1083.6GDP [trillion $] (median) 75.8 139.3 230.4 348.7 490.2 652.7 988.0 1558.3GDP [trillion $] (95% quantile) 75.8 156.5 272.1 428.0 620.7 852.6 1351.0 2244.8Temperature rise (5% quantile) [◦C] 0.9 1.0 1.3 1.6 1.8 1.9 1.9 1.8Temperature rise (median) [◦C] 0.9 1.3 1.7 2.1 2.4 2.5 2.7 3.1Temperature rise (95% quantile) [◦C] 0.9 1.5 2.0 2.4 2.6 2.9 3.4 4.9Abatement Expenditures [trillion $] 0.01 0.12 0.73 2.41 4.98 6.69 6.99 6.00Emission Control Rate 0.22 0.25 0.44 0.65 0.83 0.93 1 1

DICE GDP [trillion $] (5% quantile) 75.8 123.9 195.1 284.6 389.3 501.9 721.9 1061.4GDP [trillion $] (median) 75.8 139.1 229.9 348.4 491.3 652.1 979.9 1536.2GDP [trillion $] (95% quantile) 75.8 156.3 271.4 427.5 620.9 850.2 1335.9 2213.3Temperature rise (5% quantile) [◦C] 0.9 1.0 1.2 1.4 1.4 1.2 0.9 0.6Temperature rise (median) [◦C] 0.9 1.2 1.6 1.9 2.2 2.4 2.2 2.2Temperature rise (95% quantile) [◦C] 0.9 1.5 2.0 2.6 3.1 3.5 4.4 6.4Abatement Expenditures [trillion $] 0.05 0.24 0.82 2.20 4.91 8.45 7.74 6.13Emission Control Rate 0.20 0.32 0.54 0.62 0.81 1 1 1

Table 8: Optimal vs. DICE Abatement Policy for Level Impact. The table summarizes thesimulation results obtained by running our model (L-N) with the optimal abatement policy and with theDICE abatement policy.

the latest version of DICE. In our model, however, society optimally anticipates environmental

shocks and adjusts both the optimal abatement rate and the consumption rate. Along a path

with high optimal abatement (as a response to high temperatures), the corresponding SCC

values are significantly larger than the estimates in DICE. It is important to mention that DICE

is formulated in a purely deterministic setting. In particular the temperature dynamics are

calibrated to expected environmental outcomes, but do not take the uncertainty immanent in

the climate system into account.

To analyze these points, we run our model with the optimal abatement policy obtained from

DICE. Notice that following this policy is suboptimal in our model. The simulation results are

summarized in Table 8. It turns out that the DICE abatement policy is more stringent than the

median optimal policy. This leads to significant GDP losses, since the benefits of the DICE policy

are lower than their abatement costs. Additionally, the DICE abatement policy is insensitive

to unexpected variations in temperature, since it is determined in a deterministic model. By

contrast, the optimal abatement policy reacts to high temperatures by tightening the abatement

activities. This raises the social cost of carbon beyond the optimal value suggested by DICE.

Conversely, along paths with low abatement, society raises consumption and SCC values are

smaller. In contrast to the outcomes of following the (suboptimal) DICE policy, the variation

of optimally controlled global temperatures and in turn the variation of climate damages is

significantly smaller, while the variation of emissions is much higher.

26

6 Conclusion

This paper studies a flexible dynamic stochastic equilibrium model for optimal carbon abate-

ment. All key variables such as carbon concentration, global temperature and world GDP are

modeled as stochastic processes. Therefore, we can determine state-dependent optimal policies

and provide model-based confidence bands for all our results. We perform a sophisticated cal-

ibration of all three model components (carbon concentration, global temperature, economy).

In particular, we match the future distributions of transient climate response (TCR) and equi-

librium climate sensitivity (ECS) as provided in the report of the IPCC (2014).

We study both a level and growth rate impact of temperatures on output and combine these

two specifications with two alternative calibrations of the damage function. One calibration

suggests moderate effects of climate change even for high temperature as in Nordhaus and Sztorc

(2013), whereas the other calibration involves potentially severe damages as in Weitzman (2012).

Therefore, we can compare four different scenarios: growth-rate impact and moderate damages

(G-N), growth-rate impact and severe damages (G-W), level impact and moderate damages

(L-N), level impact and severe damages (L-W)). We find that depending on the specification of

the damage function the results can be very different. First, the social cost of carbon are similar

for frameworks with level or growth rate impact if the potential damages of global warming are

moderate. This changes significantly if damages can be severe. Then SCC is more than twice

as large for a growth rate impact than for a level impact. The results are qualitatively similar

for the optimal abatement policies.

We also document the effect of varying risk aversion and elasticity of intertemporal substitution

on our results. If damages are moderate for high temperatures, risk aversion only matters when

climate change has a level impact on output. Nevertheless, the effects are relatively small. By

contrast, the elasticity of intertemporal substitution has a significant effect for both level and

growth rate impact. If however damages are potentially severe for high temperatures, the results

are sensitive to the choice of risk aversion for both impacts. Now, SCC and optimal abatement

policy increase with risk aversion.

Finally, we find that in all scenarios the optimal abatement policies are state-dependent, but the

strength of this dependence varies across scenarios. Given a Nordhaus damage calibration, the

median results for optimal abatement policies and thus optimal emissions are similar, but the

variations are higher for the level than for the growth-rate impact. In both cases, the optimal

policies are less state-dependent than for the Weitzman damage calibration where the abatement

policies are more rigorous.

27

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30

−2 0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

(b) ECS without jumps

°C−2 0 2 4 6 8 100

0.05

0.1

0.15

0.2

0.25

0.3

(a) ECS (Benchmark)

°C

Figure 6: Equilibrium Climate Sensitivity Graph (a) shows a histogram of the simulated ECSusing the benchmark calibration. Graph (b) depicts a histogram of the ECS if we turn off the stochasticfeedback effects. The histograms are based on a simulation of 1 million sample paths.

A Global Warming Process

The starting point for our climate model is the empirically well-documented logarithmic re-

lationship between global warming and atmospheric carbon dioxide concentrations (see IPCC

(2014)). A deterministic description of this relation is

Tt = ητ log

(Y ΣtY PI

). (24)

Applying Ito’s lemma to (24) and using (2) implies

dTt =Ytητ

Y Σt

(µy(t)− αt −

1

2

Yt

Y Σtσ2y

)dt+

Ytητ

Y ΣtσydW

yt . (25)

Notice that σy is empirically negligible. In Section 3.2, we use historical carbon dioxide data

and estimate σy = 0.0078. This implies∣∣− 12 YtY Σt σ2y∣∣ ≤ 3 ·10−5, so that the term −12 YtY Σt σ2y is close

to zero. In the sequel, we thus drop this term.

Empirically, the relation between the temperature increase and carbon dioxide concentration

is not deterministic (as assumed in (24)), but noisy. This calls for an additional modification

wherefore we add two additional sources of randomness: First, we allow the temperature to

be driven by a Brownian shock that is not necessarily perfectly correlated with W y and that

potentially induces more noise than the shock in (25), which is in line with empirical evidence.

The latter means that we replace the diffusion parameter σy by στ . Second, there is empirical

evidence that the distribution of future temperat